Algèbres De Hecke Et Séries Principales Généralisées De Sp 4 ( F )

The aim of this work is to expand Bushnell and Kutzko's theory of G ‐covers [ Proc. London Math. Soc. 77 (1998) 582–634] up to a full description of the generalized principal series of the p ‐adic group Sp 4 ( F ), with p odd. We start with a Levi component M of a maximal parabolic subgroup P of G = Sp 4 ( F ) and an explicit type ( J M , τ M ) for the inertial class S in M of a supercuspidal representation of M . We compute the Hecke algebra of a G ‐cover ( J , τ) of ( J M , τ M ) constructed in our previous work [ Ann. Inst. Fourier 49 (1999) 1805–1851]: it is a convolution algebra on a Coxeter group (namely, the affine Weyl group of either U (1,1)( F ), in the case of the Siegel parabolic, or SL 2 ( F ), described explicitly by generators and relations. From this and Bushnell and Kutzko's work we derive the structure of the parabolically induced representationsi n d P G π , for π in S , and we find their discrete series subrepresentations if any, thus recovering, through the theory of G ‐covers, results previously obtained by Shahidi using different methods. The paper is written in French. 2000 Mathematical Subject Classification : 22E50, 11F70.